3 Description

F07VHF (DTBRFS) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular band system of linear equations with multiple right-hand sides AX=B or ATX=B. The routine handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of F07VHF (DTBRFS) in terms of a single right-hand side b and solution x.

Given a computed solution x, the routine computes the component-wisebackwarderrorβ. This is the size of the smallest relative perturbation in each element of A and b such that x is the exact solution of a perturbed system

A+δAx=b+δbδaij≤βaij and δbi≤βbi.

Then the routine estimates a bound for the component-wiseforwarderror in the computed solution, defined by:

6 Error Indicators and Warnings

If INFO=-i, the ith parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The bounds returned in FERR are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

8 Further Comments

A call to F07VHF (DTBRFS), for each right-hand side, involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately 2nk floating point operations (assuming n≫k).